Multipole accelerated magnetic field calculations for superconducting circuits

This paper presents an adaptive fast multipole algorithm to accelerate the computation of magnetic fields surrounding current-carrying superconducting volumes. The algorithm is based on a hierarchical tree of cubic cells and vector spherical harmonics are used to approximate the gradient of Green's function. The derivation of the multipole and local expansion coefficients are presented, including the implementation of these coefficients within FastCap's multipole algorithm. We demonstrate how the proposed algorithm can be used to calculate the magnetic fields of trapped flux and the magnetic fields surrounding type-II superconducting microstrips and compare the results to analytical solutions. The overall complexity of our multipole algorithm is found empirically to rise linearly, O(N), with N evaluation points.